|Theories may be removed if ...|
A source of speculation that has existed for years within the fandom is whether or not Aegon Targaryen survived. A Dance With Dragons reveals that the character is alive and living under the false name of Young Griff in order to hide his true identity. A number of fans believe the character to be a fraud.
Did Aegon survive?
Fans have wondered for years whether or not Aegon survived which was spurred on by George Martin not confirming that the character died in one of his communications. He did confirm that there was no doubt that Aegon's sister died. 
In A Dance With Dragons, Tyrion Lannister discovers that a youth going by the name of Young Griff is truly Aegon Targaryen in disguise. A story is given of how Aegon was switched with another baby by Varys and spirited away to be raised in secret until he could claim the Iron Throne. He fits the description of a Targaryen with purple eyes and silver blond hair that he dyes blue.
The son of Rhaegar Targaryen and Elia Martell
The circumstance that the switch occurred during of the Sack of King's Landing has brought up the question whether Varys would have had enough time to switch the children. However, it is important to know that King Aerys II had already planned the Wildfire plot, the destruction of King's Landing through wildfire, prior to the Sack. Knowledge of the Wildfire Plot could have given the spymaster enough time to find a suitable baby to switch Aegon with, giving the Targartyen dynasty a chance to survive. Additionally, Varys might have foreseen the possibility that King's Landing would, eventually, fall, and thus prepared for the situation by finding suitable child.
It may also be the case that Gregor Clegane or his party learned of the switch, so Clegane opted to destroy the boy's head to prevent a proper identification. This being so, it also became necessary to kill Elia, if she also knew of the switch.
In 300 AC, Varys murdered Kevan Lannister, who was Lord Regent at this time, in order to prevent him from clearing up the mess caused by his niece, Cersei Lannister, and to secure the campaign of the Golden Company. While Kevan was dying, Varys revealed that Aegon is the son of Rhaegar.
"Aegon has been shaped for rule before he could walk. He has been trained in arms, as befits a knight to be, but that was not the end of his education. He reads and writes, he speaks several tongues, he has studied history and law and poetry. A septa has instructed him in the mysteries of the Faith since he was old enough to understand them. He has lived with fisherfolk, worked with his hands, swum in rivers and mended nets and learned to wash his own clothes at need. He can fish and cook and bind up a wound, he knows what it is like to be hungry, to be hunted, to be afraid. Tommen has been taught that kingship is his right. Aegon knows kingship is his duty, that a king must put his people first, and live and rule for them".
Because Varys told this to a dying man, it is believed he had no reason to lie.
Not the son of Rhaegar and Elia
A prominent theory is that Varys is presenting a false Aegon Targaryen in order to place an imposter on the Iron Throne. Varys's speech to Kevan Lannister does not conclusively imply that Aegon is a Targaryen, but instead that he is simply well-qualified for the throne (based on his qualities if not by blood), as Tyrion Lannister remarks to himself " the perfect prince".
Quaithe told Daenerys Targaryen to beware "the mummer's dragon". Either this refers to Aegon being under the control of "the mummer" - Varys was a mummer in his childhood and is a master actor - or that he's a false dragon, a fake Targaryen.
At the House of the Undying, Daenerys is referred to as the Slayer of Lies while she sees 3 visions. The second vision is of A cloth dragon swayed on poles amid a cheering crowd. This could mean that she will encounter a fake Targaryen.
The fact that it should not have been possible for Varys to know in advance that Aegon would be killed in such a manner as to leave him unrecognizable is often seen as a key problem with the pisswater prince story. If the pisswater prince's face had remained intact after his death (as Rhaenys' was), no one would have been fooled. Moreover, the Valyrian features of silver-gold hair and purple eyes are rare in Westeros, making it is unlikely that the pisswater prince could have had them. Pale blond hair may pass for silver-gold if there is very little left and it's covered in blood, but Varys could not have known in advance that Aegon's whole head would be smashed. While possible explanations can be given (e.g. Varys's knowledge of Aerys's wildfire plot made him certain that the pisswater prince would become unidentifiable; or, Varys had not been planning to fool the rebels into thinking Aegon had died, but needed a substitute child in place only long enough to have sufficient time to smuggle the real Aegon out of King's Landing), part of the readers see the issue as an argument against Varys's story. In their opinions, the pisswater prince plot only makes sense retrospectively, knowing that the baby presented by Tywin Lannister was not recognizable as Aegon. They feel it seems more likely that Varys cooked up this plot after the real Aegon had died to exploit the shred of doubt that the corpse's condition had created.
As such, multiple possible theories have arisen to explain who Aegon is, summarized below.
A Blackfyre descendant
Some readers have the theory that Aegon is in truth a Blackfyre, a descendant of a legitimized branch of House Targaryen which contested the succession to the Iron Throne. Officially, the male line of the Blackfyres was wiped out when Ser Barristan Selmy killed Maelys I Blackfyre, but Aegon could be related to this House. Bookreaders have theorized that the Golden Company would never back a Targaryen after they have fought against them for generations. Illyrio told Tyrion Lannister that the Blackfyres are extinct in the male line, which could imply that there are still female-line descendants. Or that Maelys I had a wife and children, in secret. In this case, because of his age, Aegon could be his grandson. Or he is in a similar way related to another Blackfyre; Daemon I Blackfyre had seven sons and an unknown number of daughters. While three of his sons (Aegon, Aemon and Daemon presumably died without issue, at least one of Daemon's sons (Haegon) fathered issue, and several other of his children lived until adulthood. A cousin of Maelys I, another Daemon Blackfyre, lived during the same timespan as Maelys; his ancestry, as well as any possible descendants, is equally unknown.
Son of Illyrio Mopatis
Illyrio Mopatis is noted to have had a wife with Valyrian features in the past, and it may be that Aegon is her son by Illyrio. Illyrio may have even married a Blackfyre descendant who would have been a granddaughter or great-granddaughter of Daemon (whose heirs besides his first few children and his grandson Maelys are unknown). This would explain why Illyrio was seemingly eager to help Viserys and Daenerys to make their claim for the Iron Throne, yet apparently wanted nothing in return. He and Varys may have had no real intention of crowning either, but they needed Viserys's claim to build an army in advance of crowning Aegon. They may have intended to kill Viserys and Daenerys, or they may have expected them to be killed by one of Viserys's many enemies.
Just a boy with the right age and look
Shortly before his death Kevan Lannister mentioned in a Small Council meeting that Aegon isn't Aegon, just a boy as part of a plot to gain support for the Golden Company. That's also possible because in Essos, in the Free Cities, are people who descend from Valyria, with the purple eyes and the hair of silver-gold or platinum white. It is known that Targaryen kings looked for brides in the Free Cities like Lys to find right matches for their sons, especially when there were no other girls available, like sisters/cousins of House Targaryen or daughters of other Valyrian families, like House Velaryon. Varys and Illyrio Mopatis could have found a boy with the right age and look to portray him as Aegon, so that he can gain support. Varys is originally from Lys and had as spymaster still contacts to all Free Cities, including his hometown so that he could arrange such a scheme. Ironically this could mean that even if Aegon is really just a boy of right look and age, he could still have Targaryen-blood in his veins because Aerion Targaryen, who was called "Brightflame", has spent time in Lys, rode with the Second Sons and could have fathered a few children during his time there.